FORCES IN TIIE SAME PLANE. [CHAP. I. 



the direction of action of the force, first naming the point of 

 application, and then the end. 



A force due to the composition of several forces, as P 1? P 2 , P 8 , 

 we denote by P w or R^. Thus R^ denotes the resultant of 

 the forces P 1? P 2 , and P 8 . 



il. Parallelogram of Forces. If two forces, P! and Pj, 

 given in direction and intensity by the lines OP t OP 2 [Fig. 1, 

 PI. 1], have a common point of application O, the resultant 

 Rj.2 is found by the well known principle of the " parallelo- 

 gram of forces," by completing the parallelogram as indicated 

 by the dotted lines, and drawing the diagonal. OR then gives 

 the resultant of the forces P x and P.J. If this resultant acts in 

 the direction from O to R, as indicated by the arrow, it replaces 

 P! and P 2 ; that is, it produces the same effect as both forces 

 acting together. If it were taken as acting in the opposite 

 direction i.e., from O outwards, away from R it would hold 

 the forces p t arid P 2 in equilibrium. 



Now, we see at once that it is unnecessary to complete the 

 parallelogram. It is sufficient to draw from the end of the 

 force P 2 the line P 2 R in the same direction that PI acts in, and 

 make it equal and parallel to P x . The point R thus found is 

 the end of the resultant R, or is a point upon the direction of 

 the resultant prolonged through O. 



As to the direction of action of the resultant if we follow 

 round the triangle from O to P 2 and from P 2 to R and R to O 

 i.e., if we follow round in the direction of the forces the 

 direction for the resultant from R to O thus obtained is, as we 

 have already seen, the direction necessary for equilibrium. 



3. If, instead of two forces, we have three or more, as P t , P 2 , 

 P 3 , P 4 [Fig. 2] we still have the same construction. Thus com- 

 pleting the parallelogram for P! and P 2 we find R^. Complet- 

 ing the parallelogram for R t . 2 and P 8 , we find R w , and again, 

 with this and P 4 we obtain R w . Again, we see it is unneces- 

 sary to complete all the parallelograms. We have only to draw 

 lines P t R^, Ri_ 2 R^, Ri. 8 R w , parallel to the forces P 2 P 8 and 

 P 4 respectively, and equal in length to the intensities of these 

 forces, and then, no matter what may be the number of forces, 

 the line drawn from the point of beginning to the end of the 

 last Ime laid ^will give the intensity and position of the 

 resultant. As to direction, the same holds good as before. 



If the end of the last line laid off as above, should coincide 



